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Theorem elrnmpt2d 5980
Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypotheses
Ref Expression
elrnmpt2d.1 𝐹 = (𝑥𝐴𝐵)
elrnmpt2d.2 (𝜑𝐶 ∈ ran 𝐹)
Assertion
Ref Expression
elrnmpt2d (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem elrnmpt2d
StepHypRef Expression
1 elrnmpt2d.2 . 2 (𝜑𝐶 ∈ ran 𝐹)
2 elrnmpt2d.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
32elrnmpt 5972 . . 3 (𝐶 ∈ ran 𝐹 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
43ibi 267 . 2 (𝐶 ∈ ran 𝐹 → ∃𝑥𝐴 𝐶 = 𝐵)
51, 4syl 17 1 (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wrex 3068  cmpt 5231  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  ablsimpg1gend  20140  elrgspnlem4  33235
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